UDC: 517.968
STUDY OF A CLASS OF OPERATOR-DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
Zarifzoda S.K., Rozikov M.T., Bobiev M.M.
Tajik National University
Today, the theory of integro-differential equations is one of the independent areas of research, developing independently of other sections of mathematics. A detailed description of the achievements in this area before the sixties of the last century can be found in the article by M.M. Weinberg [1] (see also the bibliography in this work). The main feature of the theory of integro-differential equations is manifested in their two-dimensional nature. That is, in such equations, the unknown function appears both under the differential sign and under the integral sign. A mathematical description of the laws of development of systems containing hereditary information or dynamic memory, V. Volterra [2] (see monograph [2] and literature to it). A number of important results devoted to the study of integro-differential equations were published in the works of scientists I.N. Vekua [3], L.G. Magnaradze [4], N.A. Sidorov [5], M.V. Falaleev [6], S.S. Orlova [7] and others.
In recent years, a new theory of the class of integral equations with singular kernels was developed in the works of N. Rajabov [8], [9]. Based on these results, L.N. Rajabov [10] investigated two-dimensional integral equations of the Volterra type with singular kernels.

The methodology in the works of N. Rajabov was created and allowed to study the class of integro-differential equations with kernels of various specialties in the works [11]-[23].
Problem statement and method of its investigation. Let Γ={x: a_1≤x≤a_3 } be a set of points on the real axis, and let a_2∈ be a point Γ. With Γ_1 ва Γ_2 respectively denote the following fragments:
D_x^([a_1 a_2 ] a_3 )=(x-a_1 )(a_2-x)( a_3-x) d/dx.
In Γ_2 we include such operator:
D_x^(a_1 [a_2 a_3 ] )=(x-a_1)(x- a_2)( a_3-x)d/dx.
In the notation D_x^([a_1 a_2 ] a_3 ) we consider that x∈[a_1,a_2 ] and in the notation D_x^(a_1 [a_2 a_3 ] ) we consider that x∈[a_2,a_3 ]. In this case, the coefficients of both operators D_x^([a_1 a_2 ] a_3 ) and D_x^(a_1 [a_2 a_3 ] ) are positive.
Now we consider the following operator-differential equation in Γ_1
[D_x^([a_1 a_2 ] a_3 ) ]^2 y + MD_x^([a_1 a_2 ] a_3 ) y+Ny=f(x), (1)
where M, N are constant coefficients and f(x) is the right side of the equation and is the given function. In equation (1), the symbol [D_x^([a_1 a_2 ] a_3 ) ]^2 is understood in the following sense
[D_x^([a_1 a_2 ] a_3 ) ]^2=⏟(D_x^([a_1 a_2 ] a_3 ) )┬2 ⏟(D_x^([a_1 a_2 ] a_3 ) )┬1.
In the last writing, the operator marked with 1 is implemented first, and the operator marked with 2 is implemented after the first operator is implemented.
The inhomogeneous equation (1) corresponds to the corresponding homogeneous equation
[D_x^([a_1 a_2 ] a_3 ) ]^2 y + MD_x^([a_1 a_2 ] a_3 ) y+Ny=0 (2)

To find the solution of the homogeneous equation (2), it is necessary to know the characteristic function of the operator D_x^([a_1 a_2 ] a_3 ).
Definition 1. The function φ(x) is called the characteristic function of the operatorD_x^([a_1 a_2 ] a_3 ) if the following condition is fulfilled
D_x^([a_1 a_2 ] a_3 ) φ(x)=λφ(x),

where λ is a constant number independent of x.
It can be easily shown that the characteristic function of the operator D_x^([a_1 a_2 ] a_3 ) has the following form
φ(x)=(x-a_1 )^(λ/(a_(1 )–a_2 )(a_(1 )–a_3 ) ) (a_2-x)^(λ/(a_(2 )–a_1 )(a_(2 )–a_3 ) ) (a_3-x)^(λ/(a_(3 )–a_1 )(a_(3 )–a_2 ) ) (3)

On this basis, since the characteristic function of the operator D_x^([a_1 a_2 ] a_3 )has the type (3), then we are looking for the solution of the homogeneous equation (2) in the same type
y=〖(x-a_1)〗^(λ/((a_(1 )-a_2)(a_(1 )-a_3) )) 〖(a_2-x)〗^(λ/((a_(2 )-a_1)(a_(2 )-a_3) )) 〖(a_3-x)〗^(λ/((a_(3 )-a_1)(a_(3 )-a_2))) (4)

that is, in the type where λ is currently an unknown parameter.
For the sake of brevity, we will use the following notations in the future
λ^(a_1 )=λ/((a_(1 )-a_2)(a_(1 )-a_3) ),λ^(a_2 )=λ/((a_(2 )-a_1)(a_(2 )-a_3) ),λ^(a_3 )=λ/((a_(3 )-a_1)(a_(3 )-a_2) ),
1^(a_1 )=1/((a_(1 )-a_2)(a_(1 )-a_3) ),1^(a_2 )=1/((a_(2 )-a_1)(a_(2 )-a_3) ),1^(a_3 )=1/((a_(3 )-a_1)(a_(3 )-a_2) ).

Then formula (4) takes the following form
y=〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) (5)

We apply the operator D_x^([a_1 a_2 ] a_3 ) twice to the function (5) and obtain
D_x^([a_1 a_2 ] a_3 ) y=D_x^(a_1 [a_2 a_3 ] ) [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=
=λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ),
[D_x^([a_1 a_2 ] a_3 ) ]^2 y=D_x^([a_1 a_2 ] a_3 ) D_x^([a_1 a_2 ] a_3 ) [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=
=D_x^([a_1 a_2 ] a_3 ) [λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=
=λD_x^([a_1 a_2 ] a_3 ) [λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=
=λ∙λ[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=
=λ^2 [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ].
We put the values D_x^([a_1 a_2 ] a_3 ) y, [D_x^([a_1 a_2 ] a_3 ) ]^2 y and y itself into equation (2)
λ^2 [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]+
+Mλ[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]+
+N[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=0.
In this equation, we remove the single multiplicand from the parentheses
(λ^2+Mλ+N)[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]=0.
Here
λ^2+Mλ+N=0, (6)
that is, we derive a second-order algebraic equation with respect to the parameter λ. Equation (6) is called the characteristic equation for homogeneous equation (2).
Now, depending on the roots of the characteristic equation (6), we derive the solution of the homogeneous equation (2) in the following cases.
I. Let D=M^2-4N>0 and λ_1,λ_2 be its real and different roots in equation (6). Then the specific solutions of equation (2) are functions
y_1=(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ),
y_2=(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) )
and its general solution is a function
y=c_1 [(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_2^(a_1 ) ) (a_3-x)^(λ_3^(a_1 ) ) ]+
+c_2 [(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] (7)
where c_1,c_2 are arbitrary constants.
II. Let D=0 and λ_1=λ_2=λ be its true and identical roots in equation (6). Then special cases of equation (2) take the form
y_1=[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ],
y_2=[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]
and general solutions take the form
y=c_3 [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]+
+c_4 [〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ],
or
y=[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]×
×[c_3+c_4 ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ] (8)
where c_3,c_4 are optional constants.

III. Let D<0 in the characteristic equation (6). In this case, equation (6) has the following two complex and combined roots:
λ_1,2=(-M±√(M^2-4N))/2=(-M±√(-(4N-M^2 ) ))/2=
=(-M±√(4N-M^2 )∙√(-1))/2=(-M±√(4N-M^2 )∙i)/2==-M/2±√(4N-M^2 )/2 i.
If we include α=-M/2,β=√(4N-M^2 )/2 signs, then
λ_1,2=α±iβ.
In this case, the specific solutions of the homogeneous equation (2) take the form y_1=[〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ]×
×cos⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ],
y_2=[〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ]×
×sin⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ]
and its general solution takes the form
y=c_5 [〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ]×
×cos⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ]+
+c_6 [〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ]×
×sin⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ],
ё
y=[〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ]×
×[c_5 cos⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ]+┤
+c_6 ├ sin⁡[β ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ] ] (9)

Now we begin to find the solution of the inhomogeneous equation (1). For this, we use the method of variation of arbitrary constants.
I. If the roots of the characteristic equation (6) are real and different, we look for the solution of the inhomogeneous equation (1) in type
y=c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] (10)
where c_1 (x) and c_2 (x) are unknown functions. To find these unknown functions, we apply the operator D_x^([a_1 a_2 ] a_3 ) to the function (10)
D_x^([a_1 a_2 ] a_3 ) y=D_x^([a_1 a_2 ] a_3 ) c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+D_x^([a_1 a_2 ] a_3 ) c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]+
+λ_1 c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] +
+λ_2 c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ].
Let the optional functions c_1 (x) and c_2 (x) be such that the following condition is fulfilled
[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_1 (x)+
+[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_2 (x)=0 (11)
then
D_x^([a_1 a_2 ] a_3 ) y=λ_1 c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] +
+λ_2 c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]. (12)
We apply the operator D_x^([a_1 a_2 ] a_3 )to this equality again.
[D_x^([a_1 a_2 ] a_3 ) ]^2 y=λ_1 D_x^([a_1 a_2 ] a_3 ) c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+λ_2 D_x^([a_1 a_2 ] a_3 ) c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]+
+λ_1^2 c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] +
+λ_2^2 c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ].

We put the valuesD_x^([a_1 a_2 ] a_3 ) y ва [D_x^([a_1 a_2 ] a_3 ) ]^2 y into equation (1)
λ_1 D_x^([a_1 a_2 ] a_3 ) c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+λ_2 D_x^([a_1 a_2 ] a_3 ) c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]+
+λ_1^2 c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] +
+λ_2^2 c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]+
+Mλ_1 c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+Mλ_2 c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]
+Nc_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+Nc_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]=f(x).
We simplify the last equation
λ_1 D_x^([a_1 a_2 ] a_3 ) c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ]+
+λ_2 D_x^([a_1 a_2 ] a_3 ) c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]+
+(λ_1^2+Mλ_1+N) c_1 (x)[(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] +
+(λ_2^2+Mλ_2+N) c_2 (x)[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ]=f(x).
As if λ_1^2+Mλ_1+N=0 and λ_2^2+Mλ_2+N=0
than
λ_1 [(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_1 (x)+
+λ_2 [(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_2 (x)=f(x). (13)

Combining equations (10) and (12) to determineD_x^([a_1 a_2 ] a_3 ) c_1 (x) and D_x^([a_1 a_2 ] a_3 ) c_2 (x)we obtain the following system of algebraic equations
{█([(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_1 (x)+@+[(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_2 (x)=0,@λ_1 [(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_1 (x)+@+λ_2 [(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_2 (x)=f(x).)┤

Using Cramer's rule, we solve this system:
∆=
=|■((x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) )&(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) )@λ_1 (x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) )&λ_2 (x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) )|=
=(λ_2-λ_1 ) (x-a_1 )^(λ_(a_1)^1+λ_(a_1)^2 ) (a_2-x)^(λ_(a_1)^1+λ_(a_1)^2 ) (a_3-x)^(λ_(a_1)^1+λ_(a_1)^2 ).
∆_(c_1 (x) )=|■(0&(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) )@f(x)&λ_2 (x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) )|=
=-(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) f(x),
∆_(c_2 (x) )=|■((x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) )&0@λ_1 (x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) )&f(x) )|=
=(x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) f(x),
D_x^([a_1 a_2 ] a_3 ) c_1 (x)=∆_(c_1 (x) )/∆=
=(-(x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) f(x))/((λ_2-λ_1 ) (x-a_1 )^(λ_1^(a_1 )+λ_2^(a_1 ) ) (a_2-x)^(λ_1^(a_2 )+λ_2^(a_2 ) ) (a_3-x)^(λ_1^(a_3 )+λ_2^(a_3 ) ) )=
=(-1)/(λ_2-λ_1 )∙f(x)/((x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) ).

D_x^([a_1 a_2 ] a_3 ) c_2 (x)=∆_(c_2 (x) )/∆=
=((x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) f(x))/((λ_2-λ_1 ) (x-a_1 )^(λ_1^(a_1 )+λ_2^(a_1 ) ) (a_2-x)^(λ_1^(a_2 )+λ_2^(a_2 ) ) (a_3-x)^(λ_1^(a_3 )+λ_2^(a_3 ) ) )=
=1/(λ_2-λ_1 )∙f(x)/((x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) ).
Here
{█(d/dx c_1 (x)=-1/(λ_2-λ_1 )∙1/((x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) ) )∙f(x)/((x-a_1)(a_2-x)( a_3-x)),@d/dx c_2 (x)=1/(λ_2-λ_1 )∙1/((x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) ) )∙f(x)/(x-a_1 )(a_2-x)( a_3-x) ,)┤

That is
█(c_1 (x)=-1/(λ_2-λ_1 ) ∫_(a_1)^x▒1/((t-a_1 )^(λ_1^(a_1 ) ) (a_2-t)^(λ_1^(a_2 ) ) (a_3-t)^(λ_1^(a_3 ) ) )∙f(t)dt/((t-a_1)(a_2-t)( a_3-t))+с ̃_1 ,@c_2 (x)=1/(λ_2-λ_1 ) ∫_(a_1)^x▒1/((t-a_1 )^(λ_2^(a_1 ) ) (a_2-t)^(λ_2^(a_2 ) ) (a_3-t)^(λ_2^(a_3 ) ) )∙f(t)dt/((t-a_1)(a_2-t)( a_3-t))+с ̃_2 )
(14)
Putting the found values of c_1 (x) and c_2 (x) from (14) to (10), we obtain after simplification:
y=с ̃_1 (x-a_1 )^(λ_1^(a_1 ) ) (a_2-x)^(λ_1^(a_2 ) ) (a_3-x)^(λ_1^(a_3 ) )+
+с ̃_2 (x-a_1 )^(λ_2^(a_1 ) ) (a_2-x)^(λ_2^(a_2 ) ) (a_3-x)^(λ_2^(a_3 ) )-
-1/(λ_2-λ_1 ) ∫_(a_1)^x▒[((x-a_1)/(t-a_1 ))^(λ_1^(a_1 ) ) ((a_2-x)/(a_2-t))^(λ_1^(a_2 ) ) ((a_3-x)/(a_3-t))^(λ_1^(a_3 ) )-┤
-├ ((x-a_1)/(t-a_1 ))^(λ_2^(a_1 ) ) ((a_2-x)/(a_2-t))^(λ_2^(a_2 ) ) ((a_3-x)/(a_3-t))^(λ_2^(a_3 ) ) ] f(t)dt/(t-a_1 )(a_2-t)( a_3-t) .(15)
Thus, the formula (15) represents the general solution of the inhomogeneous equation (1) in the case that the roots of the characteristic equation (6) are true and different.
It can be seen from (15) that if max ⁡{λ_1,λ_2 }>0, then the subintegral function has a singularity at the point t=a_1. Therefore, in order to ensure the integral approximation of the right side of (15), we require the function f(t) to become zero at the point t=a_1, and its behavior is determined according to the following asymptotic formula
f(t)=o[(t-a_1 )^δ ],δ>max⁡{λ_1,λ_2 }/(a_(1 )–a_2 )(a_(1 )–a_3 ) . (16)
When the condition (16) is fulfilled, the integral of the right side of (15) is approximated, and the solution of the equation (1) is expressed in the form (15).
Thus, the following theorem was proved regarding the solvability of the inhomogeneous equation (1):
Theorem 1. Let the coefficients M and N in the operator-differential equation (1) be such that the roots of the characteristic equation (6) are real and different. Also, let the right side of the equation (1) of the function f(x) become zero at the point x=a_1 when max⁡{λ_1,λ_2 }>0, and its behavior is determined by the asymptotic formula (16). Then the inhomogeneous equation (1) is solvable, and its general solution depends on two arbitrary constants c_1, c_2 and is given by formula (15).
II. If the roots of characteristic equation (6) are true and identical, we find the solution of equation (1) in the following form:
y=c_3 (x)[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]+
+c_4 (x)[〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ]×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ],(17)
where c_3 (x),c_4 (x) are arbitrary unknown functions. To determine these functions, we proceed as before:
D_x^([a_1 a_2 ] a_3 ) y=D_x^([a_1 a_2 ] a_3 ) c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+D_x^([a_1 a_2 ] a_3 ) c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+
+λc_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+λc_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+
+c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ).

We assume that the unknown functions c_3 (x) and c_4 (x) are such that the condition
〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_3 (x)+
+〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_4 (x)=0

is fulfilled. In this case
D_x^([a_1 a_2 ] a_3 ) y=λc_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1].

We apply the operator D_x^([a_1 a_2 ] a_3 ) to both sides of this equality.
[D_x^([a_1 a_2 ] a_3 ) ]^2 y=λD_x^([a_1 a_2 ] a_3 ) c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+D_x^([a_1 a_2 ] a_3 ) c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]+
+λ^2 c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ^2 ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+2λ].

We put these found valuesD_x^([a_1 a_2 ] a_3 ) y, [D_x^([a_1 a_2 ] a_3 ) ]^2 y and the value of the function y into equation (1)
λD_x^([a_1 a_2 ] a_3 ) c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+D_x^([a_1 a_2 ] a_3 ) c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]+
+λ^2 c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ^2 ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+2λ]+
+Mλc_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+Mc_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]+
+Nc_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+Nc_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]=f(x).

This equality is easily reduced to the following form
(λ^2+Mλ+N) c_3 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+(λ^2+Mλ+N) c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+
+(2λ+M) c_4 (x) 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+
+λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_3 (x)+
+〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]∙D_x^([a_1 a_2 ] a_3 ) c_4 (x)=f(x).

Since (λ^2+Mλ+N)=0 and 2λ+M=0, we obtained the following condition
λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_3 (x) +
+〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]∙D_x^([a_1 a_2 ] a_3 ) c_4 (x)=f(x).

So, to find the unknown functions c_3 (x), c_4 (x), we derived the following system of algebraic equations:
{█(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_3 (x)+@+〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×@×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] D_x^([a_1 a_2 ] a_3 ) c_4 (x)=0,@λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_3 (x)+@+〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×@×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1] D_x^([a_1 a_2 ] a_3 ) c_4 (x)=f(x).)┤

We solve this system by Cramer's rule:
Δ=|█(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )@λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) )┤
├ █(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]@〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) [λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1] )┤|
=〖(x-a_1)〗^(〖2λ〗^(a_1 ) ) 〖(a_2-x)〗^(〖2λ〗^(a_2 ) ) 〖(a_3-x)〗^(〖2λ〗^(a_3 ) )×
×[λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1]-
-λ〖(x-a_1)〗^(〖2λ〗^(a_1 ) ) 〖(a_2-x)〗^(〖2λ〗^(a_2 ) ) 〖(a_3-x)〗^(〖2λ〗^(a_3 ) )×
×ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]=
=〖(x-a_1)〗^(〖2λ〗^(a_1 ) ) 〖(a_2-x)〗^(〖2λ〗^(a_2 ) ) 〖(a_3-x)〗^(〖2λ〗^(a_3 ) ),
Δ_(c_3 (x) )=|█(0@f(x) )┤
├ █(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]@〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) [λ ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]+1] )┤|
=-〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ]f(x),
Δ_(c_4 (x) )=|■(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )&0@λ〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )&f(x) )|=
=〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) f(x),
D_x^([a_1 a_2 ] a_3 ) c_3 (x)=∆_(c_3 (x) )/∆=
=(-〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) [ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] ]f(x))/(〖(x-a_1)〗^(〖2λ〗^(a_1 ) ) 〖(a_2-x)〗^(〖2λ〗^(a_2 ) ) 〖(a_3-x)〗^(〖2λ〗^(a_3 ) ) )
=-(ln⁡[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] f(x))/(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) )
D_x^([a_1 a_2 ] a_3 ) c_4 (x)=∆_(c_4 (x) )/∆=(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) f(x))/(〖(x-a_1)〗^(〖2λ〗^(a_1 ) ) 〖(a_2-x)〗^(〖2λ〗^(a_2 ) ) 〖(a_3-x)〗^(〖2λ〗^(a_3 ) ) )=
=1/(〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) ) ) f(x).
Here
c_3 (x)=-∫_(a_1)^x▒ln[(t-a_1 )^(1^(a_1 ) ) (a_2-t)^(1^(a_2 ) ) (a_3-t)^(1^(a_3 ) ) ]/((t-a_1 )^(λ^(a_1 ) ) (a_2-t)^(λ^(a_2 ) ) (a_3-t)^(λ^(a_3 ) ) )×
×f(t)dt/(t-a_1 )(a_2-t)( a_3-t) +c ̃_3
c_4 (x)=∫_(a_1)^x▒1/((t-a_1 )^(λ^(a_1 ) ) (a_2-t)^(λ^(a_2 ) ) (a_3-t)^(λ^(a_3 ) ) )×
×f(t)dt/((t-a_1)(a_2-t)( a_3-t))+c ̃_4.

We put these values in the formula (16).
y=c ̃_3 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )+
+c ̃_4 〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×ln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]-
-∫_(a_1)^x▒〖((x-a_1)/(t-a_1 ))^(λ_(a_1 ) ) ((a_2-x)/(a_2-t))^(λ_(a_2 ) ) ((a_3-x)/(a_3-t))^(λ_(a_3 ) ) 〗×
× ln[(t-a_1 )^(1_(a_(1 ) ) ) (a_2-t)^(1_(a_(2 ) ) ) (a_3-t)^(1_(a_1 ) ) ]∙f(x)dt/(t-a_1 )(a_2-t)( a_3-t) +
+∫_(a_1)^x▒〖((x-a_1)/(t-a_1 ))^(λ_(a_1 ) ) ((a_2-x)/(a_2-t))^(λ_(a_2 ) ) ((a_3-x)/(a_3-t))^(λ_(a_3 ) ) 〗×
×ln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ] f(x)dt/((t-a_1)(a_2-t)( a_3-t)).
We simplify this equation

y=〖(x-a_1)〗^(λ^(a_1 ) ) 〖(a_2-x)〗^(λ^(a_2 ) ) 〖(a_3-x)〗^(λ^(a_3 ) )×
×[c ̃_3+c ̃_4 ln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+∫_(a_1)^x▒〖((x-a_1)/(t-a_1 ))^(λ^(a_1 ) ) ((a_2-x)/(a_2-t))^(λ^(a_2 ) ) ((a_3-x)/(a_3-t))^(λ^(a_3 ) ) 〗×
×ln[((x-a_1)/(t-a_1 ))^(1^(a_1 ) ) ((a_2-x)/(a_2-t))^(1^(a_2 ) ) ((a_3-x)/(a_3-t))^(1^(a_2 ) ) ] f(x)dt/((t-a_1)(a_2-t)( a_3-t)).(18
Formula (18) represents the general solution of the inhomogeneous equation (1).
We note that if in (17) λ>0, then the integral on the right side at the point t=a_1 has the specificity of degree λ. Therefore, for this integral to be approximated, we require that the function f(x) becomes zero at the point x=a_1, and its behavior is determined by the following asymptotic formula:
f(x)=o[〖(x-a_1)〗^δ ],δ>λ/(a_1-a_2 )(a_1-a_3 ) . (19)

Thus, the following theorem was proved:
Theorem 2. Let the roots of the characteristic equation (6) be true and identical, and in equation (1) the function f(x) tends to zero when λ>0 at the point x=a_1, and its behavior is determined by the asymptotic formula (19) to be Then equation (1) is solvable, and its general solution is expressed with the help of formula (18).
III. Let D<0in the characteristic equation (5). In this case, we find the solution of the inhomogeneous equation (1) in the following form:
y(x)=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x) cos⁡[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]] ┤+
├ +c_6 (x) sin⁡[βln[(x-█(a@)_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]] ],(20)

where c_5 (x), c_6 (x) are unknown functions. For this, we apply the operator D_x^([a_1 a_2 ] a_3 )- to the function (20)
D_x^([a_1 a_2 ] a_3 ) y=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) c_5 (x)┤cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+├ D_x^([a_1 a_2 ] a_3 ) c_6 (x) sin⁡[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]] ]+
+ α(x-a_1 )^(α_(a_1 ) ) (a_2-x)^(α_(a_2 ) ) (a_3-x)^(α_(a_3 ) )×
×[c_5 (x)┤cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+├ c_6 (x) sin⁡[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]] ]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[-┤βc_5 (x)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+├ βc_6 (x)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]].
Let the following condition hold:
〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) ┤ c_5 (x)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+├ ├ [D_x^([a_1 a_2 ] a_3 ) ┤ c_6 (x)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]]=0.(21)
Then
D_x^([a_1 a_2 ] a_3 ) y=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]. (22)

Once again, we apply the operator D_x^([a_1 a_2 ] a_3 ) to both sides of the last equality.

[D_x^([a_1 a_2 ] a_3 ) ]^2 y=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+D_x^([a_1 a_2 ] a_3 ) c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[α^2 cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-α├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+c_6 (x)[α^2 sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ αβcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[-αβsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-β^2 cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+c_6 (x)[αβcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ β^2 sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=
=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+D_x^([a_1 a_2 ] a_3 ) c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[(α^2-β^2)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-2αβsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+c_6 (x)[(α^2-β^2)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ 2αβcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]. (23)

We put the valuesof y, D_x^([a_1 a_2 ] a_3 ) y and [D_x^([a_1 a_2 ] a_3 ) ]^2from (20), (22) and (23) into the inhomogeneous equation (1)
〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+D_x^([a_1 a_2 ] a_3 ) c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+
+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[(α^2-β^2)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-2αβsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+c_6 (x)[(α^2-β^2)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ 2αβcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+M〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+N〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤+
+├ c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]]=f(x).

We simplify the last equation
〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[D_x^([a_1 a_2 ] a_3 ) c_5 (x)[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+D_x^([a_1 a_2 ] a_3 ) c_6 (x)[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+
+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c_5 (x)[(α^2-β^2+Mα+N)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤-
-(2αβ+Mβ)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+c_6 (x)[(α^2-β^2+Mα+N)sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+├ (2αβ+Mβ)cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=f(x).
We show that the conditions
α^2-β^2+Mα+N=0
and
2αβ+Mβ=0
is performed. We noted above that α=-M/2 and β=√(4N-M^2 )/2. From here, we put these α and β values into the last equations
α^2-β^2+Mα+N=(-M/2)^2-(√(4N-M^2 )/2)^2+M∙(-M/2)+N=
=M^2/4-(4N-M^2)/4--M^2/2+N=(M^2-4N+M^2-2M^2+4N)/4=0/4=0,
2αβ+Mβ=β(2α+M)=β(2∙(-M/2)+M)=β(-M+M)=β∙0=0.

Thus, in order to find the value of the functions D_x^([a_1 a_2 ] a_3 ) c_5 (x) and D_x^([a_1 a_2 ] a_3 ) c_6 (x) we derived the following system of algebraic equations:
{█(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_5 (x)×@×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+@+〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_6 (x)×@×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]=0,@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_5 (x)×@×[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤-@-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) D_x^([a_1 a_2 ] a_3 ) c_6 (x)×@×[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+@+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=f(x).)┤

We solve this system:
Δ==|█(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×@×[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤-@-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]] )┤
├ █(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×@×[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+@+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]] )┤|=

=〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) )×
×cos⁡[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]×
×[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+
+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]-
-〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) )×
×sin⁡[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]×
×[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤-
-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=
=〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) )×
×[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤×
×sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+βcos^2 [βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]-
-αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]×
×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]+
+├ βsin^2 [βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=
=β(x-a_1 )^(〖2α〗_(a_1 ) ) (a_2-x)^(〖2α〗_(a_2 ) ) (a_3-x)^(〖2α〗_(a_3 ) )×
×[cos^2 [βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+
+├ sin^2 [βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]=
=β〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) )

Δ_(c_5 (x) )=|█(0@f(x) )┤
├ █(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×@×[αsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+@+├ βcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]] )┤|=
=-〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x)
Δ_(c_6 (x) )==|█(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]@〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×@×[αcos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤-@-├ βsin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]] )┤
├ █(0@f(x) )┤|=

=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x)
D_x^([a_1 a_2 ] a_3 ) c_5 (x)=(Δc_5 (x))/Δ=
=(-〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x))/(β〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) ) )
=-1/β×sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x)/(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ).
D_x^([a_1 a_2 ] a_3 ) c_6 (x)=(Δc_6 (x))/Δ=
=(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x))/(β〖(x-a_1)〗^(〖2α〗^(a_1 ) ) 〖(a_2-x)〗^(〖2α〗^(a_2 ) ) 〖(a_3-x)〗^(〖2α〗^(a_3 ) ) )
=1/β×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]f(x)/(〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) ) ).
We find the values of c_5 (x) and c_6 (x)
c_5 (x)=-1/β ∫_(a_1)^x▒sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]/(〖(t-a_1)〗^(α^(a_1 ) ) 〖(a_2-t)〗^(α^(a_2 ) ) 〖(a_3-t)〗^(α^(a_3 ) ) )×
×(f(t)dt)/((t-a_1 )(a_2-t)(a_3-t))+c ̃_1,
c_6 (x)=1/β ∫_(a_1)^x▒cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]/(〖(t-a_1)〗^(α^(a_1 ) ) 〖(a_2-t)〗^(α^(a_2 ) ) 〖(a_3-t)〗^(α^(a_3 ) ) )×
×(f(t)dt)/((t-a_1 )(a_2-t)(a_3-t))+c ̃_2.
We put these values in the formula (20).

y=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[[c ̃_5 cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]┤+
+├ c ̃_6 sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]-
-1/β ∫_(a_1)^x▒〖((x-a_1)/(t-a_1 ))^(λ^(a_1 ) ) ((a_2-x)/(a_2-t))^(λ^(a_2 ) ) ((a_3-x)/(a_3-t))^(λ^(a_3 ) ) 〗×
×[sin[βln[(t-a_1 )^(1_(a_1 ) ) (a_2-t)^(1_(a_2 ) ) (a_3-t)^(1^(a_3 ) ) ]]┤×
×cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]-
-sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]×
×├ cos[βln[(t-a_1 )^(1_(a_1 ) ) (a_2-t)^(1_(a_2 ) ) (a_3-t)^(1_(a_3 ) ) ]]]×
×(f(t)dt)/((t-a_1 )(a_2-t)(a_3-t)).

We simplify the resulting equation again
y=〖(x-a_1)〗^(α^(a_1 ) ) 〖(a_2-x)〗^(α^(a_2 ) ) 〖(a_3-x)〗^(α^(a_3 ) )×
×[c ̃_5 cos[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]┤+
+├ c ̃_6 sin[βln[(x-a_1 )^(1^(a_1 ) ) (a_2-x)^(1^(a_2 ) ) (a_3-x)^(1^(a_3 ) ) ]]]+
+1/β ∫_(a_1)^x▒〖((x-a_1)/(t-a_1 ))^(λ^(a_1 ) ) ((a_2-x)/(a_2-t))^(λ^(a_2 ) ) ((a_3-x)/(a_3-t))^(λ^(a_3 ) ) 〗×
×sin⁡[βln((x-a_1)/(t-a_1 ))^(1^(a_1 ) ) ((a_2-x)/(a_2-t))^(1^(a_2 ) ) ((a_3-x)/(a_3-t))^(1^(a_3 ) ) ] f(t)dt/(t-a_1 )(a_2-t)(a_3-t) . (24)

Thus, the formula (24) represents the general solution of the inhomogeneous equation (1).
In formula (24) when α>0, the integral is non-specific. Therefore, we require the function f(x) to become zero when x→a_1, and its behavior is determined by the asymptotic formula:
f(x)=o[〖(x-a_1)〗^δ ], δ>α/((a_1-a_2 )(a_1-a_3)) (25)
Thus, if the roots of the characteristic equation (6) are complex and combined, we obtained the following result regarding the solvability of the inhomogeneous equation (1):
Theorem 3. Let the roots of the characteristic equation (6) be true and the same, and when α>0 in equation (1), the function f(x) tends to zero when x→a_1, and its behavior is determined by the asymptotic formula (25). Then equation (1) is solvable, and its general solution is expressed with the help of formula (24).